What do $dz$ and $|dz|$ mean?

Traditionally (e.g. in Euler's writings), $dz=dx+i\;dy$ is an infinitely small change as $z$ moves an infinitely small distance from one point to another. If $\gamma$ is a curve and $z$ moves along the curve, then $f(z)\;dz$ is a product of a finite complex number $f(z)$ and an infinitely small complex number $dz$. The integral $\displaystyle\int_\gamma f(z)\;dz$ is the sum of infinitely many of those infinitely small quantities. None of this is logically rigorous. The role of this non-rigorous account within the rigorous account is that this is what is to be made rigorous.

The absolute value $|dz|=\sqrt{dx^2+dy^2}$ is the infinitely small distance that $z$ has moved along the curve. The integral $\displaystyle\int_\gamma dz$ is the sum of all the infinitely small changes in $z$, thus it is the final value minus the initial value. The integral $\displaystyle\int_\gamma |dz|$ is the sum of the infinitely small arc lengths, and is therefore the total arc length.

Maybe you're OK saying (in certain contexts) $dz\in \mathrm{Lin}(\mathbb C,\mathbb C)$. I wouldn't be surprised if $|dz|$ cannot be interpreted the same way, but if not, it's just a limitation on that way of interpreting it as a way of making these things rigorous.

Of course, everything Michael Hardy says is right. Maybe the following will also help.

If $f: \mathbb{C} \to \mathbb{C}$ is a smooth function, then we can talk about $df$. Here $df$ will take as input a real tangent vector to $\mathbb{C}$ and output a complex number. If you like, $df$ gives an element of $\mathrm{Hom}_{\mathbb{R}}(\mathbb{C}, \mathbb{C})$ at each point.

In particular, $dz = dx+ i dy$ is a literally true equation inside $\mathrm{Hom}_{\mathbb{R}}(\mathbb{C}, \mathbb{C})$. A function $f$ is holomorphic (aka analytic) if and only if $df$ is a complex multiple of $dz$ at every point, in which case $df = f' dz$.

Now, about $|dz|$. As Savinov Evgeny says, $|dz| = \sqrt{(dx)^2+(dy)^2}$. This is something which takes in a tangent vector to $\mathbb{C}$ and returns a positive real number, in a non-linear manner.

I don't know whether this is the point of confusion, but it is something which confused me for a long time and I have met other people with the same confusion. I came out of my first course on differential geometry thinking that $1$-forms intrinsically anti-commuted. So I thought that $dx \wedge dy$ made sense but $(dx)^2 + (dy)^2$, if it meant anything, would be zero. This simply isn't true. There is no problem in defining a quadratic form $(dx)^2+(dy)^2$ on the tangent space to your surface. For that matter, there is no problem defining something like $dx \otimes dx + dx \otimes dy + dy \otimes dy$, which takes in two tangent vectors and outputs a scalar in a way which is neither symmetric nor anti-symmetric. A course which is racing towards Stokes' theorem will emphasize the anti-symmetric case, but there is nothing wrong with the other expressions.